Answer :
Let's go through each part of the question and explain the solution:
Part 5:
We need to check if the fraction [tex]\(\frac{1}{2}\)[/tex] is equal to [tex]\(\frac{8}{16}\)[/tex].
To see if two fractions are equivalent, we can simplify them or check if their cross-products are equal. Simplifying [tex]\(\frac{8}{16}\)[/tex] gives us [tex]\(\frac{1}{2}\)[/tex]. Thus, these fractions are indeed equal.
Part 6:
We need to verify if [tex]\(\frac{9}{10}\)[/tex] is equal to [tex]\(\frac{27}{30}\)[/tex].
Simplifying [tex]\(\frac{27}{30}\)[/tex] involves dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplification results in [tex]\(\frac{9}{10}\)[/tex]. So, [tex]\(\frac{9}{10}\)[/tex] is equal to [tex]\(\frac{27}{30}\)[/tex].
Part 8:
We are checking if [tex]\(\frac{29}{50}\)[/tex] is equivalent to [tex]\(\frac{203}{10150}\)[/tex].
To determine if they are equal, their cross-products should match. However, the result indicates that these two fractions are not equal.
Part 9:
We simplify the fraction [tex]\(\frac{5}{-25}\)[/tex].
Dividing the numerator and the denominator by 5 gives [tex]\(\frac{1}{-5}\)[/tex], which simplifies to [tex]\(-0.2\)[/tex].
Part 11:
This part checks if the repeating decimal over 3 ([tex]\(0.\overline{3}\)[/tex]), which equals [tex]\(\frac{1}{3}\)[/tex], is equal to [tex]\(\frac{18}{27}\)[/tex].
When simplified, [tex]\(\frac{18}{27}\)[/tex] equals [tex]\(\frac{2}{3}\)[/tex], which is not the same as [tex]\(\frac{1}{3}\)[/tex]. Therefore, they are not equal.
Part 12:
Here, we need to find a denominator so that [tex]\(\frac{16}{25}\)[/tex] equals [tex]\(\frac{80}{x}\)[/tex].
To do this, set the fractions equal and solve for the unknown denominator:
[tex]\[
\frac{16}{25} = \frac{80}{x}
\][/tex]
By cross-multiplying, we get:
[tex]\[
16 \times x = 80 \times 25
\][/tex]
Simplifying gives [tex]\(x = \frac{80 \times 25}{16} = 125\)[/tex]. Thus, [tex]\(x\)[/tex] is 125.
So, the answers are:
1. True, [tex]\(\frac{1}{2} = \frac{8}{16}\)[/tex].
2. True, [tex]\(\frac{9}{10} = \frac{27}{30}\)[/tex].
3. False, [tex]\(\frac{29}{50} \neq \frac{203}{10150}\)[/tex].
4. [tex]\(-0.2\)[/tex], for [tex]\(\frac{5}{-25}\)[/tex].
5. False, [tex]\(\overline{3} \neq \frac{18}{27}\)[/tex].
6. The denominator should be 125 for [tex]\(\frac{16}{25} = \frac{80}{x}\)[/tex].
Part 5:
We need to check if the fraction [tex]\(\frac{1}{2}\)[/tex] is equal to [tex]\(\frac{8}{16}\)[/tex].
To see if two fractions are equivalent, we can simplify them or check if their cross-products are equal. Simplifying [tex]\(\frac{8}{16}\)[/tex] gives us [tex]\(\frac{1}{2}\)[/tex]. Thus, these fractions are indeed equal.
Part 6:
We need to verify if [tex]\(\frac{9}{10}\)[/tex] is equal to [tex]\(\frac{27}{30}\)[/tex].
Simplifying [tex]\(\frac{27}{30}\)[/tex] involves dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplification results in [tex]\(\frac{9}{10}\)[/tex]. So, [tex]\(\frac{9}{10}\)[/tex] is equal to [tex]\(\frac{27}{30}\)[/tex].
Part 8:
We are checking if [tex]\(\frac{29}{50}\)[/tex] is equivalent to [tex]\(\frac{203}{10150}\)[/tex].
To determine if they are equal, their cross-products should match. However, the result indicates that these two fractions are not equal.
Part 9:
We simplify the fraction [tex]\(\frac{5}{-25}\)[/tex].
Dividing the numerator and the denominator by 5 gives [tex]\(\frac{1}{-5}\)[/tex], which simplifies to [tex]\(-0.2\)[/tex].
Part 11:
This part checks if the repeating decimal over 3 ([tex]\(0.\overline{3}\)[/tex]), which equals [tex]\(\frac{1}{3}\)[/tex], is equal to [tex]\(\frac{18}{27}\)[/tex].
When simplified, [tex]\(\frac{18}{27}\)[/tex] equals [tex]\(\frac{2}{3}\)[/tex], which is not the same as [tex]\(\frac{1}{3}\)[/tex]. Therefore, they are not equal.
Part 12:
Here, we need to find a denominator so that [tex]\(\frac{16}{25}\)[/tex] equals [tex]\(\frac{80}{x}\)[/tex].
To do this, set the fractions equal and solve for the unknown denominator:
[tex]\[
\frac{16}{25} = \frac{80}{x}
\][/tex]
By cross-multiplying, we get:
[tex]\[
16 \times x = 80 \times 25
\][/tex]
Simplifying gives [tex]\(x = \frac{80 \times 25}{16} = 125\)[/tex]. Thus, [tex]\(x\)[/tex] is 125.
So, the answers are:
1. True, [tex]\(\frac{1}{2} = \frac{8}{16}\)[/tex].
2. True, [tex]\(\frac{9}{10} = \frac{27}{30}\)[/tex].
3. False, [tex]\(\frac{29}{50} \neq \frac{203}{10150}\)[/tex].
4. [tex]\(-0.2\)[/tex], for [tex]\(\frac{5}{-25}\)[/tex].
5. False, [tex]\(\overline{3} \neq \frac{18}{27}\)[/tex].
6. The denominator should be 125 for [tex]\(\frac{16}{25} = \frac{80}{x}\)[/tex].
